html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Cos

 http://functions.wolfram.com/01.07.21.2761.01

 Input Form

 Integrate[z^n E^(p z) Sin[b z^2]^m Cos[c z^2]^v, z] == 2^(-1 - m - v) ((1/p) ((2 Binomial[m, m/2] Binomial[v, v/2] Gamma[1 + n, (-p) z] (-1 + Mod[m, 2]) (-1 + Mod[v, 2]))/(-p)^n) + (Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[(-1)^k Binomial[m, k] (((-1)^m E^((I p^2)/(8 b k - 4 b m)) Sum[2^(-n + q) (I b (2 k - m))^(-(1/2) - n) (-p)^(n - q) (p + 2 I b (2 k - m) z)^(1 + q) ((I (p + 2 I b (2 k - m) z)^2)/( b (2 k - m)))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (p + 2 I b (2 k - m) z)^2)/(b (8 k - 4 m))], {q, 0, n}])/Sqrt[I b (2 k - m)] + (E^((I p^2)/(-8 b k + 4 b m)) Sum[2^(-n + q) (I b (-2 k + m))^(-(1/2) - n) (-p)^(n - q) (p + 2 I b (-2 k + m) z)^(1 + q) (-((I (p + 2 I b (-2 k + m) z)^ 2)/(b (2 k - m))))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I (p + 2 I b (-2 k + m) z)^2)/ (b (8 k - 4 m)))], {q, 0, n}])/Sqrt[I b (-2 k + m)]), {k, 0, Floor[(1/2) (-1 + m)]}])/I^m + Binomial[m, m/2] (-1 + Mod[m, 2]) Sum[Binomial[v, s] ((E^((I p^2)/(8 c s - 4 c v)) Sum[2^(-n + q) (-p)^(n - q) (I c (2 s - v))^(-(1/2) - n) (p + 2 I c (2 s - v) z)^(1 + q) ((I (p + 2 I c (2 s - v) z)^2)/(c (2 s - v)))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (p + 2 I c (2 s - v) z)^2)/ (c (8 s - 4 v))], {q, 0, n}])/Sqrt[I c (2 s - v)] + (E^((I p^2)/(-8 c s + 4 c v)) Sum[2^(-n + q) (-p)^(n - q) (I c (-2 s + v))^(-(1/2) - n) (p + 2 I c (-2 s + v) z)^(1 + q) ((I (p + 2 I c (-2 s + v) z)^2)/(c (-2 s + v)))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I (p + 2 I c (-2 s + v) z)^2)/( c (8 s - 4 v)))], {q, 0, n}])/Sqrt[I c (-2 s + v)]), {s, 0, Floor[(1/2) (-1 + v)]}] - Sum[Binomial[v, s] Sum[(-1)^k Binomial[m, k] ((E^((I p^2)/(-8 b k + 4 b m - 8 c s + 4 c v)) Sum[2^(-n + q) (-p)^(n - q) ((-I) (2 b k - b m + 2 c s - c v))^( -(1/2) - n) (p - 2 I (2 b k - b m + 2 c s - c v) z)^(1 + q) ((I (I p + 2 (2 b k - b m + 2 c s - c v) z)^2)/(2 b k - b m + 2 c s - c v))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (I p + 2 (2 b k - b m + 2 c s - c v) z)^2)/ (8 b k - 4 b m + 8 c s - 4 c v)], {q, 0, n}])/ Sqrt[(-I) (2 b k - b m + 2 c s - c v)] + (E^(I (m Pi + p^2/(8 b k - 4 b m - 8 c s + 4 c v))) Sum[2^(-n + q) (-p)^(n - q) (I (2 b k - b m - 2 c s + c v))^( -(1/2) - n) (-((I (I p + 2 (-2 b k + b m + 2 c s - c v) z)^2)/ (2 b k - b m - 2 c s + c v)))^((1/2) (-1 - q)) (p + 2 I (2 b k - b m - 2 c s + c v) z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, -((I (I p + 2 (-2 b k + b m + 2 c s - c v) z)^ 2)/(8 b k - 4 b m - 8 c s + 4 c v))], {q, 0, n}])/ Sqrt[I (2 b k - b m - 2 c s + c v)] + (E^((I p^2)/(-8 b k + 4 b m + 8 c s - 4 c v)) Sum[2^(-n + q) (-p)^(n - q) ((-I) (2 b k - b m - 2 c s + c v))^( -(1/2) - n) (p - 2 I (2 b k - b m - 2 c s + c v) z)^(1 + q) ((I (I p + 2 (2 b k - b m - 2 c s + c v) z)^2)/(2 b k - b m - 2 c s + c v))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (I p + 2 (2 b k - b m - 2 c s + c v) z)^2)/ (8 b k - 4 b m - 8 c s + 4 c v)], {q, 0, n}])/ Sqrt[(-I) (2 b k - b m - 2 c s + c v)] + (E^(I (m Pi + p^2/(8 b k - 4 b m + 8 c s - 4 c v))) Sum[2^(-n + q) (-p)^(n - q) (I (2 b k - b m + 2 c s - c v))^( -(1/2) - n) (p + 2 I (2 b k - b m + 2 c s - c v) z)^(1 + q) (-((I (I p + 2 (b (-2 k + m) + c (-2 s + v)) z)^2)/ (2 b k - b m + 2 c s - c v)))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I (I p + 2 (b (-2 k + m) + c (-2 s + v)) z)^2)/(8 b k - 4 b m + 8 c s - 4 c v))], {q, 0, n}])/Sqrt[I (2 b k - b m + 2 c s - c v)]), {k, 0, Floor[(1/2) (-1 + m)]}], {s, 0, Floor[(1/2) (-1 + v)]}]/ I^m) /; Element[m, Integers] && m > 0 && Element[v, Integers] && v > 0 && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]], "m"], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "]"]], "v"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m", "-", "v"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "p"], RowBox[List["(", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "p"]], ")"]], RowBox[List["-", "n"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["-", "p"]], " ", "z"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["-", "m"]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["p", "2"]]], RowBox[List[RowBox[List["8", " ", "b", " ", "k"]], "-", RowBox[List["4", " ", "b", " ", "m"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "p"]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["8", " ", "k"]], "-", RowBox[List["4", " ", "m"]]]], ")"]]]]]]], "]"]]]]]]]], ")"]], "/", RowBox[List["(", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["p", "2"]]], RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", "b", " ", "k"]], "+", RowBox[List["4", " ", "b", " ", "m"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "p"]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["8", " ", "k"]], "-", RowBox[List["4", " ", "m"]]]], ")"]]]]]]]]], "]"]]]]]]]], ")"]], "/", RowBox[List["(", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]], ")"]]]]]], ")"]]]]]]]], "+", RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["p", "2"]]], RowBox[List[RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "p"]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["8", " ", "s"]], "-", RowBox[List["4", " ", "v"]]]], ")"]]]]]]], "]"]]]]]]]], ")"]], "/", RowBox[List["(", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["p", "2"]]], RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", "c", " ", "s"]], "+", RowBox[List["4", " ", "c", " ", "v"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "p"]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["8", " ", "s"]], "-", RowBox[List["4", " ", "v"]]]], ")"]]]]]]]]], "]"]]]]]]]], ")"]], "/", RowBox[List["(", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]], ")"]]]]]], ")"]]]]]]]], "-", RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["-", "m"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["p", "2"]]], RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", "b", " ", "k"]], "+", RowBox[List["4", " ", "b", " ", "m"]], "-", RowBox[List["8", " ", "c", " ", "s"]], "+", RowBox[List["4", " ", "c", " ", "v"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "p"]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List[RowBox[List["8", " ", "b", " ", "k"]], "-", RowBox[List["4", " ", "b", " ", "m"]], "+", RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]]], "]"]]]]]]]], ")"]], "/", RowBox[List["(", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["m", " ", "\[Pi]"]], "+", FractionBox[SuperscriptBox["p", "2"], RowBox[List[RowBox[List["8", " ", "b", " ", "k"]], "-", RowBox[List["4", " ", "b", " ", "m"]], "-", RowBox[List["8", " ", "c", " ", "s"]], "+", RowBox[List["4", " ", "c", " ", "v"]]]]]]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "p"]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List[RowBox[List["8", " ", "b", " ", "k"]], "-", RowBox[List["4", " ", "b", " ", "m"]], "-", RowBox[List["8", " ", "c", " ", "s"]], "+", RowBox[List["4", " ", "c", " ", "v"]]]]]]]]], "]"]]]]]]]], ")"]], "/", RowBox[List["(", SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["p", "2"]]], RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", "b", " ", "k"]], "+", RowBox[List["4", " ", "b", " ", "m"]], "+", RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "p"]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List[RowBox[List["8", " ", "b", " ", "k"]], "-", RowBox[List["4", " ", "b", " ", "m"]], "-", RowBox[List["8", " ", "c", " ", "s"]], "+", RowBox[List["4", " ", "c", " ", "v"]]]]]]], "]"]]]]]]]], ")"]], "/", RowBox[List["(", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["m", " ", "\[Pi]"]], "+", FractionBox[SuperscriptBox["p", "2"], RowBox[List[RowBox[List["8", " ", "b", " ", "k"]], "-", RowBox[List["4", " ", "b", " ", "m"]], "+", RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "p"]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List[RowBox[List["8", " ", "b", " ", "k"]], "-", RowBox[List["4", " ", "b", " ", "m"]], "+", RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]]]]], "]"]]]]]]]], ")"]], "/", RowBox[List["(", SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]]]]], ")"]]]]]], ")"]]]]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z n p z sin m ( b z 2 ) cos v ( c z 2 ) z 2 - m - v - 1 ( 2 ( - p ) - n Γ ( n + 1 , - p z ) ( m mod 2 \$CellContext`m 2 - 1 ) ( v mod 2 \$CellContext`v 2 - 1 ) p ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] + - m ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( v mod 2 \$CellContext`v 2 - 1 ) k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 b ( 2 k - m ) ( ( - 1 ) m p 2 8 b k - 4 b m q = 0 n 2 q - n ( b ( 2 k - m ) ) - n - 1 2 ( - p ) n - q ( p + 2 b ( 2 k - m ) z ) q + 1 ( ( p + 2 b ( 2 k - m ) z ) 2 b ( 2 k - m ) ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( p + 2 b ( 2 k - m ) z ) 2 b ( 8 k - 4 m ) ) ) + 1 b ( m - 2 k ) ( p 2 4 b m - 8 b k q = 0 n 2 q - n ( b ( m - 2 k ) ) - n - 1 2 ( - p ) n - q ( p + 2 b ( m - 2 k ) z ) q + 1 ( - ( p + 2 b ( m - 2 k ) z ) 2 b ( 2 k - m ) ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , - ( p + 2 b ( m - 2 k ) z ) 2 b ( 8 k - 4 m ) ) ) ) + ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 c ( 2 s - v ) ( p 2 8 c s - 4 c v q = 0 n 2 q - n ( - p ) n - q ( c ( 2 s - v ) ) - n - 1 2 ( p + 2 c ( 2 s - v ) z ) q + 1 ( ( p + 2 c ( 2 s - v ) z ) 2 c ( 2 s - v ) ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( p + 2 c ( 2 s - v ) z ) 2 c ( 8 s - 4 v ) ) ) + 1 c ( v - 2 s ) ( p 2 4 c v - 8 c s q = 0 n 2 q - n ( - p ) n - q ( c ( v - 2 s ) ) - n - 1 2 ( p + 2 c ( v - 2 s ) z ) q + 1 ( ( p + 2 c ( v - 2 s ) z ) 2 c ( v - 2 s ) ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , - ( p + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) ) ) - - m s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( p 2 - 8 b k + 4 b m + 8 c s - 4 c v q = 0 n 2 q - n ( - p ) n - q ( - ( 2 b k - b m - 2 c s + c v ) ) - n - 1 2 ( p - 2 ( 2 b k - b m - 2 c s + c v ) z ) q + 1 ( ( p + 2 ( 2 b k - b m - 2 c s + c v ) z ) 2 2 b k - b m - 2 c s + c v ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( p + 2 ( 2 b k - b m - 2 c s + c v ) z ) 2 8 b k - 4 b m - 8 c s + 4 c v ) ) / ( - ( 2 b k - b m - 2 c s + c v ) ) + ( ( p 2 8 b k - 4 b m - 8 c s + 4 c v + m π ) q = 0 n 2 q - n ( - p ) n - q ( ( 2 b k - b m - 2 c s + c v ) ) - n - 1 2 ( - ( p + 2 ( - 2 b k + b m + 2 c s - c v ) z ) 2 2 b k - b m - 2 c s + c v ) 1 2 ( - q - 1 ) ( p + 2 ( 2 b k - b m - 2 c s + c v ) z ) q + 1 ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , - ( p + 2 ( - 2 b k + b m + 2 c s - c v ) z ) 2 8 b k - 4 b m - 8 c s + 4 c v ) ) / ( ( 2 b k - b m - 2 c s + c v ) ) + ( p 2 - 8 b k + 4 b m - 8 c s + 4 c v q = 0 n 2 q - n ( - p ) n - q ( - ( 2 b k - b m + 2 c s - c v ) ) - n - 1 2 ( p - 2 ( 2 b k - b m + 2 c s - c v ) z ) q + 1 ( ( p + 2 ( 2 b k - b m + 2 c s - c v ) z ) 2 2 b k - b m + 2 c s - c v ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( p + 2 ( 2 b k - b m + 2 c s - c v ) z ) 2 8 b k - 4 b m + 8 c s - 4 c v ) ) / ( - ( 2 b k - b m + 2 c s - c v ) ) + ( ( p 2 8 b k - 4 b m + 8 c s - 4 c v + m π ) q = 0 n 2 q - n ( - p ) n - q ( ( 2 b k - b m + 2 c s - c v ) ) - n - 1 2 ( p + 2 ( 2 b k - b m + 2 c s - c v ) z ) q + 1 ( - ( p + 2 ( b ( m - 2 k ) + c ( v - 2 s ) ) z ) 2 2 b k - b m + 2 c s - c v ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , - ( p + 2 ( b ( m - 2 k ) + c ( v - 2 s ) ) z ) 2 8 b k - 4 b m + 8 c s - 4 c v ) ) / ( ( 2 b k - b m + 2 c s - c v ) ) ) ) /; m + v + n Condition z z n p z b z 2 m c z 2 v 2 -1 m -1 v -1 2 -1 p -1 n Gamma n 1 -1 p z \$CellContext`m 2 -1 \$CellContext`v 2 -1 p -1 Binomial m m 2 -1 Binomial v v 2 -1 -1 m Binomial v v 2 -1 \$CellContext`v 2 -1 k 0 m -1 2 -1 -1 k Binomial m k 1 b 2 k -1 m 1 2 -1 -1 m p 2 8 b k -1 4 b m -1 q 0 n 2 q -1 n b 2 k -1 m -1 n -1 1 2 -1 p n -1 q p 2 b 2 k -1 m z q 1 p 2 b 2 k -1 m z 2 b 2 k -1 m -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 p 2 b 2 k -1 m z 2 b 8 k -1 4 m -1 1 b m -1 2 k 1 2 -1 p 2 4 b m -1 8 b k -1 q 0 n 2 q -1 n b m -1 2 k -1 n -1 1 2 -1 p n -1 q p 2 b m -1 2 k z q 1 -1 p 2 b m -1 2 k z 2 b 2 k -1 m -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 p 2 b m -1 2 k z 2 b 8 k -1 4 m -1 Binomial m m 2 -1 \$CellContext`m 2 -1 s 0 v -1 2 -1 Binomial v s 1 c 2 s -1 v 1 2 -1 p 2 8 c s -1 4 c v -1 q 0 n 2 q -1 n -1 p n -1 q c 2 s -1 v -1 n -1 1 2 p 2 c 2 s -1 v z q 1 p 2 c 2 s -1 v z 2 c 2 s -1 v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 p 2 c 2 s -1 v z 2 c 8 s -1 4 v -1 1 c v -1 2 s 1 2 -1 p 2 4 c v -1 8 c s -1 q 0 n 2 q -1 n -1 p n -1 q c v -1 2 s -1 n -1 1 2 p 2 c v -1 2 s z q 1 p 2 c v -1 2 s z 2 c v -1 2 s -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 p 2 c v -1 2 s z 2 c 8 s -1 4 v -1 -1 -1 m s 0 v -1 2 -1 Binomial v s k 0 m -1 2 -1 -1 k Binomial m k p 2 -8 b k 4 b m 8 c s -1 4 c v -1 q 0 n 2 q -1 n -1 p n -1 q -1 2 b k -1 b m -1 2 c s c v -1 n -1 1 2 p -1 2 2 b k -1 b m -1 2 c s c v z q 1 p 2 2 b k -1 b m -1 2 c s c v z 2 2 b k -1 b m -1 2 c s c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 p 2 2 b k -1 b m -1 2 c s c v z 2 8 b k -1 4 b m -1 8 c s 4 c v -1 -1 2 b k -1 b m -1 2 c s c v 1 2 -1 p 2 8 b k -1 4 b m -1 8 c s 4 c v -1 m q 0 n 2 q -1 n -1 p n -1 q 2 b k -1 b m -1 2 c s c v -1 n -1 1 2 -1 p 2 -2 b k b m 2 c s -1 c v z 2 2 b k -1 b m -1 2 c s c v -1 1 2 -1 q -1 p 2 2 b k -1 b m -1 2 c s c v z q 1 Binomial n q Gamma q 1 2 -1 -1 p 2 -2 b k b m 2 c s -1 c v z 2 8 b k -1 4 b m -1 8 c s 4 c v -1 2 b k -1 b m -1 2 c s c v 1 2 -1 p 2 -8 b k 4 b m -1 8 c s 4 c v -1 q 0 n 2 q -1 n -1 p n -1 q -1 2 b k -1 b m 2 c s -1 c v -1 n -1 1 2 p -1 2 2 b k -1 b m 2 c s -1 c v z q 1 p 2 2 b k -1 b m 2 c s -1 c v z 2 2 b k -1 b m 2 c s -1 c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 p 2 2 b k -1 b m 2 c s -1 c v z 2 8 b k -1 4 b m 8 c s -1 4 c v -1 -1 2 b k -1 b m 2 c s -1 c v 1 2 -1 p 2 8 b k -1 4 b m 8 c s -1 4 c v -1 m q 0 n 2 q -1 n -1 p n -1 q 2 b k -1 b m 2 c s -1 c v -1 n -1 1 2 p 2 2 b k -1 b m 2 c s -1 c v z q 1 -1 p 2 b m -1 2 k c v -1 2 s z 2 2 b k -1 b m 2 c s -1 c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 p 2 b m -1 2 k c v -1 2 s z 2 8 b k -1 4 b m 8 c s -1 4 c v -1 2 b k -1 b m 2 c s -1 c v 1 2 -1 m SuperPlus v SuperPlus n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "n_"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", "z_"]]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]], "]"]], "m_"], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]], "]"]], "v_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m", "-", "v"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "p"]], ")"]], RowBox[List["-", "n"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["-", "p"]], " ", "z"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], "p"], "+", RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["-", "m"]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["p", "2"]]], RowBox[List[RowBox[List["8", " ", "b", " ", "k"]], "-", RowBox[List["4", " ", "b", " ", "m"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "p"]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["8", " ", "k"]], "-", RowBox[List["4", " ", "m"]]]], ")"]]]]]]], "]"]]]]]]]], SqrtBox[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["p", "2"]]], RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", "b", " ", "k"]], "+", RowBox[List["4", " ", "b", " ", "m"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "p"]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["8", " ", "k"]], "-", RowBox[List["4", " ", "m"]]]], ")"]]]]]]]]], "]"]]]]]]]], SqrtBox[RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]]]], ")"]]]]]]]], "+", RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["p", "2"]]], RowBox[List[RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "p"]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["8", " ", "s"]], "-", RowBox[List["4", " ", "v"]]]], ")"]]]]]]], "]"]]]]]]]], SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["p", "2"]]], RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", "c", " ", "s"]], "+", RowBox[List["4", " ", "c", " ", "v"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "p"]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["8", " ", "s"]], "-", RowBox[List["4", " ", "v"]]]], ")"]]]]]]]]], "]"]]]]]]]], SqrtBox[RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]], ")"]]]]]]]], "-", RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["-", "m"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["p", "2"]]], RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", "b", " ", "k"]], "+", RowBox[List["4", " ", "b", " ", "m"]], "-", RowBox[List["8", " ", "c", " ", "s"]], "+", RowBox[List["4", " ", "c", " ", "v"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "p"]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List[RowBox[List["8", " ", "b", " ", "k"]], "-", RowBox[List["4", " ", "b", " ", "m"]], "+", RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]]], "]"]]]]]]]], SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["m", " ", "\[Pi]"]], "+", FractionBox[SuperscriptBox["p", "2"], RowBox[List[RowBox[List["8", " ", "b", " ", "k"]], "-", RowBox[List["4", " ", "b", " ", "m"]], "-", RowBox[List["8", " ", "c", " ", "s"]], "+", RowBox[List["4", " ", "c", " ", "v"]]]]]]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "p"]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List[RowBox[List["8", " ", "b", " ", "k"]], "-", RowBox[List["4", " ", "b", " ", "m"]], "-", RowBox[List["8", " ", "c", " ", "s"]], "+", RowBox[List["4", " ", "c", " ", "v"]]]]]]]]], "]"]]]]]]]], SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["p", "2"]]], RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", "b", " ", "k"]], "+", RowBox[List["4", " ", "b", " ", "m"]], "+", RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "p"]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List[RowBox[List["8", " ", "b", " ", "k"]], "-", RowBox[List["4", " ", "b", " ", "m"]], "-", RowBox[List["8", " ", "c", " ", "s"]], "+", RowBox[List["4", " ", "c", " ", "v"]]]]]]], "]"]]]]]]]], SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["m", " ", "\[Pi]"]], "+", FractionBox[SuperscriptBox["p", "2"], RowBox[List[RowBox[List["8", " ", "b", " ", "k"]], "-", RowBox[List["4", " ", "b", " ", "m"]], "+", RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "p"]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List[RowBox[List["8", " ", "b", " ", "k"]], "-", RowBox[List["4", " ", "b", " ", "m"]], "+", RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]]]]], "]"]]]]]]]], SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]]]]]]]], ")"]]]]]]]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]], "&&", RowBox[List["v", "\[Element]", "Integers"]], "&&", RowBox[List["v", ">", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2002-12-18